In systems using MIMO such as LTE, the secondary station (or User Equipment or UE) can give the primary station (or base station or eNB) feedback on the downlink channel state. This can partly comprise an index to a precoding matrix selected from a codebook of matrices. Alternatively, as proposed for LTE-A, the precoder is defined by a pair of indices, each for one of two codebooks, where the precoder is derived from the matrix multiplication of the two matrices. In this case there could be more than one particular type of “matrix multiplication” that could be applied.
Typically a precoding matrix is defined such that the coefficients in column of the matrix represent the precoding coefficients applied the each transmit antenna for a given spatial channel.
A constraint on the codebook design to ensure that CQI calculation can be consistent with equal power per transmit antenna, at least with a subset of codebook entries, was proposed. This is intended to support full power amplifier (PA) utilization, where the same total output power is required for each antenna:[ww*]mm=κ, m=1, . . . ,NT 
Where W is the overall precoder, and NT is the number of transmit antennas.
Moreover, it is possible that at least a subset of codebook entries should also have orthogonal columns with unit norm (i.e. corresponding to unitary precoding).
In the RANI discussion of codebook design the desirability of a restricted alphabet (e.g. QPSK (Quadrature Phase Shift Keying), 8-PSK or 16-PSK) for precoding coefficients has been mentioned. One advantage of using an alphabet based on higher order M-PSK (e.g. M=8 or 16) is that it can better match the channel characteristics that low order M-PSK (e.g. M=4). Restricting strictly to M-PSK would ensure that requirements for both full PA utilization and unit norm are automatically met for all codebook entries. There may also be some reduction in computational complexity with restricted alphabets, but it is not clear how significant this consideration would be in practice. However, it is of interest to examine what other alphabets could be beneficial (e.g. whether different amplitude values should be allowed within a precoder). In principle, an ideal precoder, even with power balancing between antennas, would require an unconstrained alphabet, but we focus here on limited alphabets.
We could consider the optimum allocation of power among the precoding coefficients as analogous to the “water filling” problem. It is well known that “constant power water filling” (i.e. allocating either zero or uniform power) is quite close to the optimal solution, assuming that unused power can be re-allocated elsewhere. This suggests that adding the possibility of “zero” to an M-PSK alphabet will achieve much of the potential benefit available from an alphabet with different amplitudes.
The general principle of setting some elements of the precoder to zero is already known.